Explain 5 theorems of bond pricing.

 Bond prices move inversely to interest rates.

 Holding maturity constant, a decrease in rates will raise bond prices on a percentage basis more than a corresponding increase in rates will lower bond prices. All things being equal, bond price volatility is an increasing function of maturity. A related principle regarding maturity is as follows: the percentage price change that occurs as a result of the direct relationship between a bond’s maturity and its price volatility increases at a decreasing rate as time to maturity increases. In addition to the maturity effect, the change in the price of a bond as a result of a change in interest rates depends on the coupon rate of the bond.Explain how to control interest rate riskby using duration. A decline (rise) in interest rates will cause a rise (decline) in bond prices, with the most volatility in bond prices occurring in longer maturity bonds and bonds with low coupons. As a stock risk is usually proxied by its beta, which is a measure of the stock sensitivity to market movements, bond price risk is most often measured in terms of the bond interest-rate sensitivity, or duration. This is a one-dimensional measure of the bond’s sensitivity to interest-rate movements. There are actually three related notions of duration: Macaulay duration, $duration and modified duration. Macaulay duration, often simply called duration, is defined as a weighted average maturity for the portfolio. The duration of a bond or bond portfolio is the investment horizon such that investors with that horizon will not care if interest rates drop or rise as long as changes are small, as capital gain risk is offset by reinvestment risk on the period. $duration and modified duration are used to compute, respectively, the absolute P&L and the relative P&L of a bond portfolio for a small change in the yield to maturity. $duration also provides us with a convenient hedging strategy: to offset the risks related to a small change in the level of the yield curve, one should optimally invest in a hedging asset a proportion equal to the opposite of the ratio of the $duration of the bond portfolio to be hedged by the $duration of the hedging instrument. Properties of the Different Duration MeasuresThe main properties of duration, modified duration and $duration measures are as follows: The duration of a zero-coupon bond equals its time to maturity. Holding the maturity and the YTM of a bond constant, the bond’s duration (modified duration or $duration) is higher when the coupon rate is lower. Holding the coupon rate and the YTM of a bond constant, its duration (or modified duration) increases with its time to maturity as $duration decreases. Holding other factors constant, the duration (or modified duration) of a coupon bond is higher as $duration is lower when the bond’s YTM is lower. The duration of a perpetual bond that delivers an annual coupon c over an unlimited horizon and with a YTM equal to y is (1 + y)/y. Another convenient property of duration is that it is a linear operator. In other words, the duration of a portfolio P invested in n bonds denominated in the same currency with weights wi is the weighted average of each bond’s duration